python - Scipy: efficiently generate a series of integration (integral function) -

i have function, want integral function, this:

that is, instead of getting single integration value @ point x, need values @ multiple points.

for example:

let's want range @ (-20,20)

def f(x):     return x**2  x_vals  = np.arange(-20, 21, 1) y_vals =[integrate.nquad(f, [[0, x_val]]) x_val in x_vals ]  plt.plot(x_vals, y_vals,'-', color = 'r') 

the problem

in example code give above, each point, integration done scratch. in real code, f(x) pretty complex, , it's multiple integration, running time slow(scipy: speed integration when doing whole surface?).

i'm wondering if there way of efficient generating phi(x), @ giving range.

my thoughs:

the integration value @ point phi(20) calucation phi(19), , phi(19) phi(18) , on. when phi(20), in reality series of (-20,-19,-18,-17 ... 18,19,20). except didn't save value.

so i'm thinking, possible create save points integrate function, when passes save point, value saved , continues next point. therefore, single process toward 20, value @ (-20,-19,-18,-17 ... 18,19,20)

one could implement strategy outlined integrating on short intervals (between consecutive x-values) , taking cumulative sum of results. this:

import numpy np import scipy.integrate si def f(x):     return x**2 x_vals = np.arange(-20, 21, 1) pieces = [si.quad(f, x_vals[i], x_vals[i+1])[0] in range(len(x_vals)-1)] y_vals = np.cumsum([0] + pieces) 

here pieces integrals on short intervals, summed produce y-values. written, code outputs function 0 @ beginning of range of integration -20. 1 can, of course, subtract y-value corresponds x=0 in order have same normalization on plot.

that said, split-and-sum process unnecessary. when find indefinite integral of f, solving differential equation f' = f. , scipy has built-in method that, odeint. use it:

import numpy np import scipy.integrate si def f(x):     return x**2 x_vals = np.arange(-20, 21, 1) y_vals = si.odeint(lambda y,x: f(x), 0, x_vals) 

the output essential identical first version (within tiny computational errors), less code. reason using lambda y,x: f(x) first argument of odeint must function taking 2 arguments, right-hand side of equation y' = f(y, x).